Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f_*\mathcal{O}_X$ and $R^2f_*\mathcal{O}_X$ are both locally free $\mathcal{O}_S$modules. Can $\mathrm{dim}_{k(s)}H^0\big(X_s, \Omega^1_{X_{s}/k(s)}\big)$ and $\mathrm{dim}_{k(s)}H^1\big(X_s, \Omega^1_{X_{s}/k(s)}\big)$ depend on the closed point $s\in S$?

1$\begingroup$ The Euler characteristic of $\Omega^1_{X_s/k(s)}$ is constant (Hartshorne III 12). $\endgroup$ – Piotr Achinger Jul 2 '20 at 18:17

2$\begingroup$ You changed the question after I answered it. Please indicate the extent of such substantial edits in the future. $\endgroup$ – Piotr Achinger Jul 2 '20 at 19:22
Yes. An Enriques surface with classical reduction at $p=2$ gives such an example. See Illusie "Complexe de de RhamWitt et cohomologie cristalline" Prop. II 7.3.8(b), p. 658.

1$\begingroup$ TBH, I don't know offhand if they exist over $\mathbf{Z}[1/N]$ specifically, not just some $\mathcal{O}_K[1/N]$ with $N$ odd and $K$ a number field unramified at $2$. $\endgroup$ – Piotr Achinger Jul 2 '20 at 20:29

1$\begingroup$ You can always use Weil restriction to produce an example over Q[1/N] whose geometric fibers are products of Enriques surfaces. Anyway, there are parameter spaces if Enriques surfaces that are rational over Spec Z. $\endgroup$ – Jason Starr Jul 3 '20 at 15:34